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Security Limits of Mining Before Validation in Nakamoto Consensus

Yifan Zhou, Jiang Xiao, Kaihua Qin

arXiv:2609.29222v1cs.CRcs.DC

TL;DR

Mining before validation creates a security gap because rejecting an invalid block also discards honest descendants, a loss missed by analyses that assume instantaneous validation. The paper models bounded network and validation delays, proves conditions for security, and shows that a carrier attack yields a matching asymptotic boundary in the fully decentralized regime.

  • Problem

    Prior analyses largely treat validation as instantaneous, although rejecting an invalid block can discard the honest work built on it.

    Source: §1 · p. 2

  • Method

    The paper models Nakamoto consensus with bounded network delay and a separate load-independent validation-time bound, then analyzes fully validated chains and constructs a carrier attack.

    Source: §2 · p. 3, §1 · p. 3

  • Results

    When λaτ > 1, a carrier attack removes a target block at any fixed finite initial confirmation depth with probability one; security guarantees hold if λa < λh/(1 + λh∆) and λaτ < 1, with a matching asymptotic boundary in the fully decentralized regime.

    Source: §1 · p. 3, §4 · p. 8, §7 · p. 16

  • Takeaways & Limitations

    Under a validation-time bound independent of processing load, validation latency imposes a security limit beyond the constraint from network delay.

    Source: §7 · p. 16

    The model excludes validation queues and processing limits, and the tight threshold for mixed honest mining policies remains open.

    Source: §7 · p. 16

Abstract

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Mining before validation allows miners to extend a newly received block before completing its validity checks, giving them a head start in the race for the next block reward. This head start, however, comes with a security risk: rejecting one invalid block also discards the honest work built on it, an effect missed when validation is treated as instantaneous. We quantify this risk in a model of Nakamoto consensus with bounded network delay and a validation-time bound independent of processing load. We establish an explicit threshold on adversarial mining power below which honest miners' fully validated chains continue to grow and agree on a stable history with high probability over any fixed observation period. We also construct an attack that repeatedly draws honest mining onto invalid branches. When the adversary produces more than one block on average during the allowed validation time, the attack can eventually remove a target block at any fixed initial confirmation depth. Combined with ordinary private mining, this attack yields a matching asymptotic bound in the fully decentralized regime, where each honest miner has negligible mining power. The results show how validation latency limits the security of mining before validation, beyond the constraint imposed by network delay.

1 Introduction

Prior security analyses compare honest and adversarial chain progress while largely treating validation as instantaneous. This paper analyzes mining before validation, proves security guarantees under explicit conditions, and constructs an attack that matches the asymptotic boundary in the fully decentralized regime.

1 Introduction

  • Earlier analyses compare honest and adversarial progress and account for network-delay losses, but largely assume instantaneous block validation.Those analyses use progress comparisons and concentration bounds to establish high-probability guarantees.Source: §1 · pp. 1–2
  • An invalid adversarial carrier can divert honest mining and later erase its descendants, so invalid-branch work does not advance a fully validated chain.The proof separates work on valid ancestry from work lost to invalid ancestry and bounds usable adversarial stock.Source: §1 · p. 2
  • The model establishes high-probability chain growth, common prefix, block liveness, and mining-time persistence when λa < λh/(1 + λh∆) and λaτ < 1.The guarantees cover every admissible adversarial strategy from genesis over any fixed finite observation horizon.Source: §1 · p. 2
  • When λaτ > 1, a carrier attack can remove a target block at any prescribed finite initial confirmation depth with probability one.Combined with ordinary private mining, the attack matches the sufficient resilience bound asymptotically when each honest miner's share is negligible.Source: §1 · p. 3

2 Model and Security Definitions

The model extends continuous-time Nakamoto consensus with bounded network delay and validation latency, allowing honest miners to extend pending blocks. Security is evaluated on fully validated chains through growth, agreement, persistence, and liveness properties.

Model

  • Mining successes are independent Poisson processes with total rate λ, adversarial rate λa = βλ, and honest rate λh = (1 − β)λ; honest miners may have any finite power distribution.Honest blocks are broadcast immediately and reach all honest miners within ∆; the adversary may withhold blocks and causally schedule their delivery and validation.Source: §2 · p. 3

Validity and validation time

  • Only adversarial blocks may be invalid, and each received block's validity is learned within τ, even with pending ancestor validations.The validation oracle is load-independent, excludes validation queues and unavailable payloads, and permits worst-case schedules that need not be realizable by a concrete implementation.Source: §2 · p. 4

Eligible chains

  • Honest miners extend longest optimistic eligible chains while validation is pending, but security is measured on their separately selected longest fully validated strict chains.When a block is rejected, its descendant subtree is removed from the eligible set.Source: §2 · p. 4

Security properties

  • Chain growth requires positive growth over sufficiently long intervals, while common prefix requires chains to agree after pruning their last k blocks.Mining-time confirmation is defined using block creation times, and properties are evaluated through a finite observation horizon.Source: §2 · p. 5
  • Mining-time persistence keeps a confirmed prefix in every later honest strict chain, while block liveness requires each qualifying window to contain a block that remains in every later chain.The persistence and liveness definitions use confirmation duration or window σ; permanence extends to all future times.Source: §2 · p. 5

3 Honest Progress and Revocation under SPV Mining

Network delay makes some concurrent honest mining redundant, while mining before validation adds a separate loss: rejecting an invalid ancestor can revoke honest descendants. The section frames progress as counted chain advancement and explains how carriers undo that progress.

3.1 Honest progress

  • Honest progress discounts latency-induced concurrency, counting one additional chain level per unit; its asymptotic rate is r⋆ = λh/(1 + λh∆).Concurrent blocks are grouped into one counted unit per propagation window, and the rate is tight when individual honest mining power is negligible.Source: §3.1 · pp. 5–6, Figure 1 · p. 6
  • When all blocks are valid, r⋆ > λa gives honest progress a rate advantage, while common-prefix violations require a competing branch to replace a suffix longer than k blocks.A private fork needs one adversarial block to match one unit of honest progress.Source: §3.1 · p. 6

3.2 Revocation under SPV mining

  • One invalid carrier can revoke several honest descendants without building an equally long valid adversarial branch.Rejecting the carrier removes its subtree; in the illustrated execution, revoked units leave strict-chain height unchanged.Source: §3 · p. 5, §3.2 · pp. 6–7, Figure 2 · p. 7

4 Security Analysis

The security analysis accounts for honest mining diverted to invalid ancestry and adversarial blocks that remain usable across intervals. Under explicit bounds on adversarial mining, it establishes high-probability chain-growth and history-agreement guarantees over a finite observation horizon.

4 Security Analysis

  • When λa < λh/(1 + λh∆) and λaτ < 1, all four guarantees hold with high probability against every admissible strategy from genesis.The guarantees are chain growth, block liveness, mining-time persistence, and common prefix.Source: §4 · p. 8

4.1 Effective time and interval loss

  • Effective honest mining time counts mining on objectively valid ancestry; interval losses account for invalid work and usable adversarial stock present at the interval's start.An invalid block can attract mining for at most τ time units after receipt, while the initial stock contributes a separate loss.Source: §4.1 · pp. 8–9

4.2 Uniform stock and chain growth

  • The analysis bounds usable unpublished stock uniformly over the observation horizon and proves honest chain growth under the stated adversarial-power conditions.The stock bound is S(u) ≤ CSK; chain growth holds for sufficiently long intervals with high probability.Source: §4.2 · pp. 9–10

4.3 Shared adversarial budget

  • The shared-budget comparison has a positive progress coefficient exactly when λa < r and λaτ < 1.It compares remaining honest progress with valid competing adversarial work while accounting for carrier losses.Source: §4.3 · p. 11

4.4 Permanent blocks and security guarantees

  • A sufficiently long interval contains an honest block permanent by its end, and a sufficiently confirmed block's prefix then persists in every future honest strict chain.These liveness and persistence guarantees hold with high probability under the stated adversarial-power conditions.Source: §4.4 · p. 11
  • Common prefix holds simultaneously over the observation horizon, but taking T → ∞ does not yield a fixed confirmation window for an infinite ledger.The required window grows with K.Source: §4.4 · pp. 12–13

5 The Carrier Attack and Resilience Upper Bound

The carrier attack uses validation windows to divert honest mining onto invalid ancestry, then publishes a growing private fork to remove a target block. Combined with ordinary private mining, it yields an asymptotic resilience upper bound matching the sufficient bound in the fully decentralized regime.

5.1 Strategy and queue dynamics

  • During each validation window, honest miners extend invalid ancestry, and rejection makes those descendants ineligible for the valid chain.The adversary releases another carrier when available, or waits for honest miners to resume valid work and mine a new carrier.Source: §5.1 · p. 13
  • When βλτ > 1, the attack removes a target block from every honest strict chain with probability one, at any prescribed finite initial confirmation depth.The result concerns success strictly above the boundary; it does not establish success at equality or a finite-horizon error exponent.Source: §5.1 · p. 13, §5 · p. 13
  • A carrier queue has positive probability of never emptying when its branching-process mean cτ exceeds one; repeated restarts make sustained suppression almost sure.Each window consumes one carrier and adds new carriers; after a failed busy period, the strategy waits for a new carrier and restarts.Source: §5.1 · p. 14
  • During sustained suppression, the valid public height stays fixed while the private fork grows, eventually becoming longer and removing b at a rejection boundary.The attack assigns positive mining power to the private fork, which excludes b.Source: §5.1 · pp. 13–14

5.2 Comparison with ordinary private mining

  • The asymptotic resilience upper bound is min{βpa(λ∆), 1/(λτ)}; it matches the sufficient threshold in the fully decentralized regime.The two boundaries coincide when βpa(λ∆)λτ = 1; the security guarantees hold strictly below the displayed boundary, and behavior at equality is unsettled.Source: §5.2 · pp. 14–15

6 Related Work

Related work studies Nakamoto consensus security, participation assumptions, processing and validation, pre-validation mining, and incentives. This paper distinguishes its model by allowing extension before validation completes while bounding validation latency independently of network delay.

Consensus security

  • Prior consensus analyses study security under synchrony or bounded network delay, including asymptotic resilience and finite-confirmation safety; ordinary private-chain attack thresholds are tight in the fully decentralized regime.A bait-and-switch attack is proved optimal in a particular bounded-delay model.Source: §6 · p. 15

Resources and participation

  • Permissionless-consensus models distinguish resource assumptions from participation assumptions, while this paper fixes aggregate mining rates to isolate delayed validation.Some eligible participants in stake-based protocols need not be online [26].Source: §6 · p. 15

Processing and validation

  • Bounded-processing models study how queues and processing policies affect security, while Tuxedo permits validation to lag by a bounded number of blocks.Here, miners may extend a received block before validation completes, with a load-independent validation-latency bound.Source: §6 · p. 15

Mining before validation

  • Cao et al. study distraction-chain double spending with headers mined before payload receipt; this paper assumes payload arrival and studies honest work discarded when invalid ancestry is rejected.Manifoldchain [8] uses multiple verified and unverified parents in a sharded protocol; this paper retains a single-parent longest-chain rule.Source: §6 · p. 15

Incentives

  • Prior incentive work examines validation costs, mining policies, and rewards, but this paper takes honest mining policy as given and does not establish equilibrium.Blockchain denial-of-service work studies incentives under block-data withholding.Source: §6 · p. 15

7 Conclusion

The conclusion identifies validation latency as an additional security constraint for mining before validation. The appendix bounds invalid-branch losses by tracking adversarial stock and effective honest mining time.

7 Conclusion

  • When honest miners extend unvalidated blocks, validation latency can lower the adversarial-power threshold for breaking consensus; when λaτ > 1, carrier attacks can remove any fixed-depth target.Combined with ordinary private mining, the attack gives a matching asymptotic upper boundary when each honest miner’s share is negligible.Source: §7 · p. 16
  • Mixed-policy security and attack bounds depend on each policy’s share of honest mining power, while tight thresholds for mixtures and validation queues remain open.Source: §7 · p. 16

A Proofs of Theorems in Sections 4 and 5

  • The stock proof contracts adversarial stock across comparisons, accounts for intervening peaks, and chooses an allowance that bounds stock uniformly over the observation interval with high probability.Its replacement bound uses max{λa/r, λaτ} < 1, with λa < r < ρ.Source: A Proofs of Theorems in Sections 4 and 5 · p. 19

A.1 Interval loss with explicit stock

  • From genesis, lost effective honest mining time is bounded by τ times the number of adversarial blocks mined on invalid ancestry.The interval-loss argument groups wasted work by first-invalid root; each root causes at most τ units of loss.Source: §A.1 · pp. 20, 22

A.2 Height growth in effective time

The section bounds growth of public valid height in effective honest mining time, then transfers that bound to miners’ fully validated chains after a propagation-and-validation lag.

A.2 Height growth in effective time

  • For every r < ρ, public valid height grows at rate r in effective honest mining time over all intervals in [0, T], with probability at least 1 − e^-κ.Here ρ = λh/(1 + λh∆), and the bound holds under every admissible strategy.Source: §A.2 · p. 22
  • The proof selects effective honest successes at least ∆ apart; each selected block extends a longest optimistic chain with valid ancestry and raises height by at least one.The waiting periods exclude at most ∆[Q_u(t) + 1] effective time.Source: §A.2 · pp. 22–23
  • Fully validated chain heights inherit the public-height bound after lag D = ∆ + τ, using the load-independent validation-time bound.A valid chain known to one honest miner reaches all honest miners within ∆ and completes validation within another τ.Source: §A.2 · p. 24

A.3 Uniform stock and effective time

This section bounds usable unpublished stock by eliminating old hidden blocks and contracting replacement stock, showing that it returns to a fixed bound in controlled time. It uses this bound to establish chain growth and construct clean leads that produce permanent honest blocks, supporting liveness, persistence, and common prefix.

A.3 Uniform stock and effective time

  • Repeated stock checks contract initial unpublished stock up to an additive fluctuation allowance and yield a uniform stock bound over the observation interval.The contraction factor is strictly below one; the checks begin at genesis, and their failure probabilities are summed over the number of checks before the horizon.Source: §A.3 · pp. 28–30

A.4 Chain growth on arbitrary intervals

  • For any γ < ρ(1−λaτ), the uniform stock and adversarial-count estimates establish chain growth on arbitrary intervals, simultaneously for honest miners.The adversarial production bound covers all interval endpoints and combines an enlarged-rate term with an additive allowance.Source: §A.4 · pp. 30–31

A.5 Return to bounded stock

  • The security proof first returns usable stock to a fixed bound, enabling a fixed mining pattern to succeed with probability bounded away from zero uniformly over starting histories.Lemma 9 controls the return time’s exponential moment in terms of starting stock, even when a stock comparison fails.Source: §A.5 · p. 32
  • Returning usable stock to a fixed bound enables clean-lead retries; a lead whose comparison never fails becomes permanent, and the resulting bounds yield liveness, persistence, and common prefix.The construction repeats after finite failures and obtains an exponential waiting-time bound for a permanent block.Source: §A.5 · pp. 32–33
  • After any finite stopping time u, Lemma 9 ensures a finite v≥u with S(v)≤m and an exponential-moment return bound, uniform over admissible strategies and histories.The bound controls both the return duration and stock at the return time.Source: §A.5 · pp. 33, 34, 36

A.6 Permanent honest blocks

This section establishes permanence and common-prefix guarantees for honest blocks by constructing blocks with a clean lead and showing that some become permanent over sufficiently long observation windows. It proves the required relation for k = k0 throughout t2 ≤ T and extends the result to every integer k ≥ k0 after additional pruning.

A.6 Permanent honest blocks

  • A clean lead of L requires zero usable stock, a D-long public-height plateau, and every valid chain omitting the honest block—including hidden branches—to be at least L blocks shorter.After any finite stopping time, the construction can produce such a block; its repeated-attempt probability bound is uniform when attempts begin with stock at most m.Source: §A.6 · pp. 36–37
  • From a clean lead, if the comparison against competing valid work never reaches the initial lead, the block remains in every strict chain thereafter.The comparison treats new valid adversarial blocks as competing work and charges invalid ancestry through lost effective time; finite failure has an exponential tail in elapsed time plus stock.Source: §A.6 · p. 38
  • After every finite stopping time, the repeated reset-and-comparison construction produces an honest block that belongs to every strict chain from its permanence time onward.A failed comparison triggers another reset; its failure probability is at most 1/4, so a comparison that never crosses eventually occurs almost surely.Source: §A.6 · pp. 40–41

A.7 Block liveness and mining-time persistence

  • With probability at least 1 − e^−κ, every interval in [0,T] of length at least 2W, where W = CWK, contains an honest block mined there and permanent by the interval’s end.This all-window guarantee yields liveness for u ≥ 2CWK and supports persistence of every σ-confirmed block when σ ≥ 2CWK.Source: §A.7 · pp. 41–42

A.8 From mining-time persistence to common prefix

  • With probability at least 1−e^−κ, pruning the last k0=⌈CcpK⌉ blocks from i(t1), where t1≤T, leaves the prefix ending at B in every Cstr.Here Ccp=(1+log 2)(2λCpers+CN).Source: §A.8 · p. 43
  • Pruning additional blocks leaves a prefix of the retained chain, so the same event that proves the relation for k = k0 also proves it for every integer k ≥ k0.The proof establishes the relation for every t2 ≥ t1, including t2 ≤ T.Source: §A.8 · p. 43

A.9 Eventual success of the carrier attack

When the adversary produces more than one block on average during the validation time, carrier busy periods can freeze valid public-chain growth while a private fork grows. Repeated trials make this attack eventually remove a target block at any prescribed finite confirmation depth.

A.9 Eventual success of the carrier attack

  • When λaτ > 1, a carrier busy period has positive probability of continuing indefinitely and freezing the valid public height.During each window, honest miners extend the carrier's invalid ancestry; rejection and replacement by another queued carrier keep the valid public height fixed.Source: §A.9 · p. 44
  • The carrier queue is explored as a family tree whose independent offspring counts are the carrier successes in successive validation-length windows.Its extinction probability is the smallest fixed point of the offspring generating function; when cτ > 1, that probability is below one.Source: §A.9 · pp. 44–45
  • Repeated busy-period trials eventually produce an infinite period almost surely, while the private fork continues gaining valid blocks.The frozen public height is finite, so the private fork eventually exceeds it and reaches a qualifying rejection boundary.Source: §A.9 · pp. 45–46
  • At the qualifying rejection boundary, publishing the longer valid private chain makes every honest strict chain switch to a branch omitting the target block.The argument applies to every prescribed finite confirmation depth because it requires only an initial shared chain of finite height.Source: §A.9 · p. 46

B Mixed Honest Mining Policies

The section extends the security analysis to fixed mixtures of miners who extend only validated blocks and miners who also extend blocks awaiting validation. It gives sufficient high-probability security guarantees for these mixtures and develops weighted-loss and height-growth bounds.

B Mixed Honest Mining Policies

  • The mixture fixes a fraction p of honest mining power to optimistic mining, while strict miners extend only fully validated chains.Security is assessed on every honest miner’s strict chain, regardless of that miner’s policy.Source: B Mixed Honest Mining Policies · p. 46
  • For every γ < ρp(1 − λa pτ), the mixed-policy model gives high-probability guarantees for honest mining progress, block liveness, mining-time persistence, and common prefix.The guarantees hold simultaneously over a fixed observation horizon with probability at least 1 − e^−κ.Source: B Mixed Honest Mining Policies · p. 47
  • The guarantee constants are uniform over finite miner counts, power distributions realizing p, and admissible adversarial strategies.They depend only on the listed model parameters.Source: B Mixed Honest Mining Policies · p. 47
  • For mixed policies with 0 < p < 1, the sufficient security bound is not shown to match an attack threshold.When pτ = 0, the bound’s second term is infinite.Source: B Mixed Honest Mining Policies · p. 47

B.1 Weighted loss and height growth

  • Weighted-loss accounting charges invalid-ancestry work to optimistic miners and bounds effective-time losses using their share p.The per-root loss is bounded by pτ, while boundary losses are bounded by bp.Source: B Mixed Honest Mining Policies · p. 46, §B.1 · pp. 47–48
  • For every r < ρp, honest-chain height growth admits a uniform high-probability lower bound over the observation horizon.The proof selects permitted honest successes on valid ancestry, using delay Δ for optimistic miners and D for strict miners.Source: §B.1 · p. 48

B.2 Stock, permanent blocks, and security

The proof controls usable adversarial stock and accounts for honest work lost to invalid ancestry. These bounds support high-probability chain growth, persistence, and common-prefix guarantees under the stated rate condition.

Permanent blocks

  • The physical lag used in the clean-lead construction is D, retaining propagation and validation delay rather than adding hp to network delay.The construction uses bphys = 2∆+τ.Source: §B.2 · pp. 49, 50, 52

Security guarantees

  • When λa hp < 1, the stock and progress bounds yield high-probability liveness, persistence, and common-prefix guarantees over the fixed observation horizon.The liveness argument gives a permanent honest block in every sufficiently long interval, while mining-time confirmation and a total-mining bound establish persistence and common prefix.Source: §B.2 · pp. 50, 54

Stock and progress

  • For 0 ≤ u ≤ t ≤ T, effective progress satisfies A(t) − A(u) ≥ (t − u) − hpZbad(u,t) − CAK, subtracting bad-ancestry loss and CAK.If λa = 0, stock is zero and A(t) = t; if hp = 0, d = M = 1 and A(t) = t, even with positive physical validation delay.Source: §B.2 · p. 50

Permanent blocks

  • A bounded-stock checkpoint-and-retry construction produces a clean lead that becomes permanent with an exponentially bounded completion time.The clean state requires zero usable stock, a public-height plateau of length D, and an L-block lead over valid branches omitting the new honest block.Source: §B.2 · pp. 52–54

Persistence

  • A mining-time confirmation gap of at least 2W makes a block an ancestor of a permanent honest block, preserving its prefix in every later honest strict chain.This yields persistence from the permanent-block guarantee.Source: §B.2 · p. 54

Common prefix

  • Bounding mining successes in the confirmation window converts persistence into common prefix after pruning k0 blocks and their descendants.The component guarantees are combined to obtain probability at least 1−e^−κ.Source: §B.2 · p. 54

B.3 Attack scope and unresolved tightness

A private fork can remove a target block despite any finite initial confirmation depth when it outgrows the public chain. The attack’s scope depends on validation scheduling, and tightness for an interior mixture remains unresolved.

Private-fork attack

  • If λa > rval(p), a valid private fork eventually overtakes the public chain and removes a target block at any prescribed finite confirmation depth.The attack delays validation of honest blocks by τ, including at their creators; equality in the rate comparison is inconclusive.Source: §B.3 · pp. 55–56

Attack scope

  • The attack’s renewal rate relies on the validation oracle allowing delayed validation at a block’s creator; immediate self-validation would change the calculation.The sufficient-security proof does not require this particular completion schedule.Source: §B.3 · p. 56

Unresolved tightness

  • The immediate-delivery attack does not match the mixed height lower bound, and tightness for an interior mixture remains unresolved.The comparison does not establish that the sufficient security bound is loose; another attack might match it.Source: §B.3 · p. 56

Attack scope

  • For ∆=0, λh=1, τ=10, p=0.01, and λa=0.2, the proposed conditions hold but rval(0.01)≈0.09596, so private mining succeeds.This is a counterexample to keeping the original height rate while scaling only the carrier-loss coefficient by p.Source: §B.3 · p. 56

B.4 Incorporating network delay into the attack

The network-delay attack bounds public-chain growth by comparing it with a virtual height that removes miners’ earlier access to their own blocks. When the private fork’s rate exceeds that bound, it eventually removes a target block at any fixed finite confirmation depth.

Attack bound

  • If λa > rnet(p)+ε∆, a private-fork attack removes a target block at any prescribed finite initial confirmation depth.The result holds for sufficiently decentralized finite honest-miner distributions realizing p.Source: §B.4 · pp. 57, 59

Public-height analysis

  • The virtual height grows almost surely at rate rnet(p), accounting for network delay and miners’ own earlier access to their blocks.The construction uses delivery delay ∆ for other honest miners and bounds actual public height by virtual height plus exceptional successes.Source: §B.4 · pp. 56, 58

Attack bound

  • For fixed honest power distributions, choosing between the two delivery schedules yields an attack whenever the adversarial share exceeds the corresponding public-height rate.The network-dependent bound becomes sufficient for all sufficiently decentralized finite distributions under the stated strict inequality.Source: §B.4 · pp. 58–59
  • When ∆>0, the network-delay attack gives a strictly stronger asymptotic upper bound than immediate delivery; when ∆=0, it recovers the immediate-delivery rate.At p=0 the bound matches the sufficient bound in the fully decentralized limit, while for 0<p<1 and τ>0 tightness remains unresolved.Source: §B.4 · p. 59

B.5 Endpoints and numerical illustration

The section identifies endpoint cases and illustrates the gap between sufficient resilience bounds and attack bounds. The numerical values are analytical bounds, not established exact thresholds.

Endpoint cases

  • At p=1 the sufficient bound recovers the all-optimistic security bound, while at p=0 validation can still delay valid-chain growth although invalid carriers waste no honest work.When τ=0, mining policies coincide and the bound reduces to the network-only bound.Source: §B.5 · p. 59

Numerical illustration

  • At p=0.5, λτ=4, and λ∆=0, the sufficient bound is approximately 0.292893, while the attack succeeds above approximately 0.367987.The intervening range remains unresolved.Source: §B.5 · p. 59
  • At p=0.5, λτ=4, and λ∆=1, the sufficient bound is approximately 0.232408 and the fully decentralized attack bound is approximately 0.288092.With 100 equal-power honest miners, half optimistic and half strict, the finite-population attack bound including ε∆ is approximately 0.292144.Source: §B.5 · p. 60
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